Pilot Based Time Delay Estimation for KSP-OFDM Systems in a Multipath Fading Environment

Transcription

1 Pilot Base Time Delay Estimation for KSP-OFDM Systems in a Multipath Faing Environment Dieter Van Welen, Freerik Simoens, Heii Steenam an Marc Moeneclaey TELI Department, Ghent University Sint-Pietersnieuwstraat 41, 9 Gent, Belgium {mvwele,fsimoens,hs,mm}@telin.ugent.be Abstract We propose two time elay estimators for known symbol paing KSP orthogonal frequency ivision multiplexing OFDM in a multipath faing environment. Both estimators make use of pilot symbols in the guar interval an known pilot carriers an take the frequency selectivity of the channel into account. The performance of the estimators is illustrate by means of simulation results for the mean square error MSE an the bit error rate BER. There is a egraation in performance compare with a receiver with perfect synchronization, especially for high E s /, but KSP-OFDM systems with the propose estimators outperform a cyclic prefix OFDM system with the time elay estimator from [1]. I. ITRODUCTIO The number of wire an wireless services has increase a lot uring the last years. This increase has create the nee for a technique that combines high ata rates with a high reliability. Orthogonal frequency ivision multiplexing OFDM is a strong caniate as it is a flexible technique that can support high ata rates, an is able to combat frequency selective channels [2]. These avantageous properties have mae OFDM a hot research topic an the OFDM technique has alreay been applie in various stanars like igital auio broacasting DAB [3], igital vieo broacasting DVB [4], in moems for igital subscriber lines xdsl [5], in wireless local area networks WLA [6],... An OFDM system can be efficiently implemente by the usage of fast Fourier transforms FFT, which is a great avantage. Before the transmission, an inverse FFT IFFT is applie to the information to be transmitte, in orer to convert the ata that are moulate in the frequency omain on the ifferent carriers into a time omain signal. Further, a guar interval is inserte to avoi inter block interference IBI between successively transmitte OFDM blocks. In the literature, there exist ifferent types of guar intervals. The two most popular guar interval techniques are the cyclic prefix CP an the zero paing ZP techniques [7]. In the cyclic prefix technique, the guar interval is transmitte before each OFDM block an consists of the last samples of the OFDM block. In ZP-OFDM, the guar interval is fille with zeros, i.e. uring the guar interval no signal is transmitte. In this paper however, we will consier a thir guar interval technique, i.e. the known symbol paing KSP technique [8]. In this technique, the guar interval is fille with known samples or pilots. Synchronization of the OFDM receiver with the OFDM transmitter requires to fin the starting point of the OFDM symbol: time offsets can cause inter carrier interference ICI an IBI [9], [1]. For CP-OFDM, several time elay estimation algorithms have been propose in the literature. The authors of [1] erive the maximum likelihoo ML estimator for a time elay in the presence of aitive white Gaussian noise AWG. The reunancy of the cyclic prefix an pilot symbols on the carriers are exploite. The blin estimator of [11] is a special case of the previous estimator an only makes use of the correlation of the cyclic prefix an the last samples of the transmitte OFDM block. A time elay estimator that makes use of a specially esigne training symbol is propose in [12] for the AWG channel. However, as it oes not employ all available information, the estimator is suboptimal. In [13], the ML time elay estimator is erive in the case of ispersive channels uner the assumption of perfect channel knowlege. The estimator uses the cyclic prefix only. However, as it is in practice very ifficult to obtain a channel estimate without knowlege about the time elay, the performance of this estimator can be seen as a lower boun on the performance of an estimator which oes not assume any knowlege about the channel. Common to the time synchronization algorithms propose for CP-OFDM is the non-negligible egraation cause by the resiual timing error at high signal-tonoise ratios SR in the presence of a faing channel. In [14], it is shown that CP-OFDM an KSP-OFDM have essentially the same performance when the guar interval length is much smaller than the number of carriers. As this is the case in all practical situations, it motivate us to consier the timing synchronization problem for KSP-OFDM, where the pilots are sprea both in the time an the frequency omain. To our knowlege, no research has been one about time elay estimation algorithms for KSP-OFDM. Both the pilot symbols in the guar interval an the pilot symbols on the pilot carriers are exploite by our estimator. The performance of the propose estimator is compare with the estimator for CP-OFDM from [1] in terms of the mean square error MSE of the time elay estimate, an in terms of the bit error rate BER.

2 a transmitter T block i 1 block i block i+1 νt t block i = 1 block i = observation vector block i = 1 b receiver + Fig. 2. Definition of the receive signal vector observation interval Fig. 1. Time-omain signal of a KSP-OFDM block a transmitte signal b receive signal an observation interval II. SYSTEM MODEL Consier a KSP-OFDM system with carriers an a guar interval of length ν. M is efine as the total number of transmitte pilot symbols of which ν are transmitte uring the guar interval an M ν on the carriers. On the ifferent carriers, we transmit blocks of symbols a i = a i,..., a i 1 T consisting of M ν pilot symbols enote as b c = b c,..., b c M ν 1 T an M ata symbols enote as a i = a i T,..., ai M 1. The guar interval consists of ν pilot symbols enote as b g = b g,..., b g ν 1 T. We efine [ E s as the transmitte energy per symbol: E s = E a i n 2] [ = E b g k 2]. The transmitte symbol vector a i is moulate on the ifferent carriers using the -point IFFT. The guar interval is inserte after the IFFT outputs. The samples of the transmitte time omain signal s i = s i,..., s i 1 T are given by F s i = H a i 1 where F enotes the FFT matrix with elements F k,l = 1 j2π kl e ; k,l =,..., 1. Figure 1 shows the time omain signal. We efine the vectors s p an s i as s p = F pb c 2 s i = F a i 3 where F p consists of the M ν columns of F H which correspon to the pilot carriers an F is given by the M columns of F H that correspon to the ata carriers. So s p an s i can be seen as the pilot an ata signal in the time omain respectively. We efine b as the total transmitte pilot signal, so b collects the contribution from the pilot carriers an the pilot symbols in the guar interval s p b = +ν b. 4 g b g t The samples s i are transmitte over a frequency selective channel with an impulse response of length L enote as h = h,..., hl 1 T. In orer to avoi inter block interference, the length of the guar interval ν is chosen so that the guar interval excees the uration of the channel impulse response: ν L 1. The receiver takes a block of samples r = r,..., r2+l 3 T. Every transmitte OFDM block, which has a uration of samples, contributes to + L 1 successive samples of the receive signal after transmission over a channel with an impulse response of L samples. The vector r contains the total contribution from only one OFDM block along with partial contributions from ajacent blocks because of its length. We assume that this block has the inex i = without loss of generality. The starting point of this block in the receive signal vector r is not known an has to be estimate see figure 2. For the etection of the ata symbols transmitte in block i, we take the receive samples from the observation interval corresponing to block i as can be seen in figure 1. The contributions from the pilot symbols of the guar intervals ark gray areas on figure 1 are first subtracte from the receive signal. The resulting system can be seen as a ZP-OFDM system. ow for ata etection in a ZP-OFDM system [7], the last ν samples of the observation interval are ae to the first ν samples of the OFDM symbol see figure 1b. The resulting block of samples is then applie to the FFT. Finally per carrier symbol etection is performe. III. TIME DELAY ESTIMATIO In this section we erive the estimator for starting from the joint likelihoo function of an h for the observation r. We rop the block inex i = for notational convenience. To keep things simple, we assume that r only contains noise besies the contribution of the consiere transmitte OFDM block s 1. We efine r as the subvector of r that collects the contributions from s: r = r,..., r + +L 2 T. Because of the alreay mentione assumption, the vector r can be written as r = Hs+w 5 where s is efine in 1 with i =, H is the +L 1 Toeplitz channel matrix whose entries are efine as H l:l+l 1,l = h; l =,..., + ν 1 an w = w,..., w + +L 2 T is 1 We only use this assumption to erive the estimator, for the simulations we will consier a continuous transmission of OFDM blocks.

3 the noise vector, where wk is white aitive Gaussian noise with variance an zero mean. The contribution of the useful signal in 5 can be written as the sum of the contribution of the ata symbols an the pilot symbols: Hs = Bh+Ah 6 where B an A are the +L 1 L Toeplitz matrices with respective entries B l:l++ν 1,l = b an A l:l+ 1,l = s ; l =,..., L 1. The istribution of the receive signal vector r given, the channel impulse response h, an the ata symbol vector a is given by pr, h, a = { C exp 1 k 1 rk ν+L 2 k= ++ν+l 1 rk 2 }. { exp 1 [r B+Ah] H [r B+Ah] where C is some irrelevant constant. This expression still epens on the unknown ata symbols a an has to be average over the unknown ata symbols in orer to be useful for our estimation problem. This averaging is rather complicate so we have to simplify 7 first. For small values of x, expx can be approximate by the first two terms of its Taylor series, i.e. expx 1+x for x 1. So for low E s /, expression 7 can be approximate by pr, h, a = C C k 1 rk ν+L 2 k= ++ν+l 1 } rk 2 7 C [r B+Ah] H [r B+Ah]. 8 Averaging 8 over the unknown ata symbols is easy now as we only nee to compute the averages of A an A H A: E[A] = an E [ A H A ] = R A See appenix for the computation of R A. This yiels for pr, h pr, h = { C 1 1 [ r H r r H Bh h H B H ] r 1 h H B H B+R A h }. 9 The ML estimates of an h can be obtaine by maximizing 9 with respect to an h. The estimate of h given is obtaine by eriving 9 with respect to h an results in ĥ = B H B+R A 1 B H r 1 When we substitute this estimate of h in 9 we obtain the function Γ 1 which only epens on : Γ 1 = 1 r H B B H B+R A 1 B H r. 11 The estimate of is then given by ˆ = argmax {Γ 1 }. 12 A secon estimator can be obtaine by totally neglecting the contributions of the unknown ata symbols in 8. This means that we neglect A in 8 an R A in 9. In that case, the estimate of h given is given by ĥ = B H B 1 B H r 13 an the estimate of is then given by with ˆ = argmax {Γ 2 } 14 Γ 2 = 1 r H B B H B 1 B H r. 15 Although we erive the joint estimate of h an in this algorithm, only the estimate for is use. Inee, the estimate for h will perform baly at high E s /, as the contributions from the ata symbols in 8 an 9 have been either neglecte or replace by their means, resulting in an error floor in the MSE of h an the BER see [15] an [14]. The erivation of the estimate of h is only neee to remove its contribution from 9 in orer to obtain a simple expression for the estimate of. For channel estimation, better estimators are available in the literature, e.g. [16], [17], having better performance at high E s / than the estimators 1 an 13. If we take a closer look at 11 an 15, we see that the functions Γ 1 an Γ 2 compute the correlation between the receive signal an the pilot vector b at L successive time instants as can be seen from the matrix prouct B H r : B H 1 r l = r + l + ks p k + ν 1 r + l + + kb g k 16 where l =,..., L 1. Both the estimators 12 an 14 try to fin the ˆ that maximizes a function of the L successive correlations between the receive signal an the pilot vector. IV. SIMULATIO RESULTS In this section the performance of our time elay estimators is evaluate by means of simulations. We compare the performance of the estimators with the ML time elay estimation algorithm for CP-OFDM from [1]. We consier = 124 carriers an a guar interval of length ν = 1 for KSP-OFDM an CP-OFDM respectively. To make a fair 2 comparison between CP- OFDM an KSP-OFDM, we assume that the number of pilot symbols transmitte on the carriers in the CP- OFDM signal is equal to M ν. The transmitte symbols consist of ranomly generate QPSK symbols. Although we erive the estimator for uner the assumption that only one OFDM block is transmitte, we simulate a 2 By taking, ν an the number M ν of pilot carriers the same for both CP-OFDM an KSP-OFDM, we obtain the same ata throughput an, assuming perfect synchronization an channel knowlege, essentially the same BER.

4 1 6 KSP OFDM 1.45 MSE KSP OFDM 2 CP OFDM p ˆk k E s / B ˆ Fig. 3. MSE results for a frequency selective channel, L = 5, = 124, ν = 1, M = 2 Fig. 5. Histogram of the time elay estimation error for the CP-OFDM estimator, E s / = 2 B, 1 pilot carriers Perfect Synchronization.9.9 KSP OFDM KSP OFDM 2 CP OFDM.6.6 p ˆk k.5.4 p ˆk k.5.4 BER ˆ ˆ E s / B Fig. 4. Histogram of the time elay estimation error for the KSP-OFDM estimator 1 left an 2 right, E s / = 2 B, M = 2 continuous transmission of OFDM symbols. As we want to focus on the impact of time elay estimation errors, it is assume for the simulation of the BER that possible phase rotations of the symbol constellation, cause by time elay estimation errors, are perfectly compensate an that the channel is perfectly estimate after the time elay estimation. For KSP-OFDM, these assumptions mean that the contributions from the pilot symbols from the guar interval can be perfectly remove from the receive signal. In the figures an in the accompanying text, the KSP-OFDM estimator from 12 which takes the unknown ata symbols in to account, is calle KSP- OFDM estimator 1, while the estimator from 14 which totally neglects the contributions from the unknown ata symbols, is calle KSP-OFDM estimator 2. The performance of the estimators in a ispersive channel is shown in figures 3-6. We consier a frequency selective Rayleigh faing channel consisting of L = 5 channel taps. Figure 3 shows the results for the MSE on the time elay estimate. The KSP-OFDM estimators Fig. 6. BER results for a frequency selective channel, L = 5, = 124, ν = 1, M = 2 outperform the estimator for CP-OFDM as coul be expecte: our estimators take the ispersive nature of the channel into account while the estimator from [1] was esigne for an AWG channel an so this estimator is not robust to a ispersive channel as oppose to our estimator. The KSP-OFDM estimator 2 outperforms the first KSP-OFDM estimator for higher E s /. Figures 4 an 5 show a histogram of the estimation error ˆ for the KSP-OFDM estimators an the CP- OFDM estimator respectively for E s / = 2 B an 1 pilot carriers M = 2. The first KSP-OFDM estimator fins the true in more than 8% of all simulate cases. The secon KSP-OFDM estimator performs even better an fins the real in more than 9% of all simulate cases. For both KSP-OFDM estimators, the estimation error ˆ is smaller than or equal to 2 samples in more than 99% of all simulate cases. The performance of the CP-OFDM estimator is much worse: the true is almost never foun an less than 1% of all cases results in ˆ 2 samples. The BER results for a ispersive channel are shown in

5 figure 6. The BER curves confirm the results from the other figures. We see that KSP-OFDM systems with the propose estimators exhibit a lower BER than the CP- OFDM system with the time elay estimator from [1]. The performance of receivers with the consiere estimators is close to a receiver with perfect synchronization for low to mile high E s /. For higher E s /, the KSP- OFDM systems will also exhibit an error floor for the BER but CP-OFDM has a significantly higher error floor. The error floors of the propose estimators are cause by the assumptions mae in the erivation of these estimator, i.e. that only one OFDM symbol is transmitte whereas in the simulations continuous transmission is consiere, an by assuming that the ata symbols can be neglecte or replace by their averages. KSP-OFDM estimator 2 results in a lower error floor than KSP-OFDM estimator 1, so totally neglecting the contribution of the unknown ata symbols for the estimation of the time elay gives better results than averaging first over the unknown ata symbols. V. COCLUSIO We have erive two time elay estimators for KSP- OFDM in multipath faing environments. Both estimators are base on the correlation between the receive signal an the pilot symbols in the guar interval an the correlation between the receive signal an the time omain contribution from the pilot carriers. The first estimator is erive after averaging the likelihoo function of the receive signal over the unknown ata symbols. The secon estimator just neglects the contribution of the unknown ata symbols. We compare the propose time elay estimators with the ML time elay estimator for a CP- OFDM system [1] in terms of MSE an BER. The KSP- OFDM systems with our time elay estimators outperform the consiere CP-OFDM system., as they result in a lower BER. The KSP-OFDM estimator which neglects the unknown ata symbols, gives better performance than the estimator which averages the likelihoo function of the receive signal first over the unknown ata symbols. APPEDIX In this appenix we compute R A which is the average of A H A. ote that A H A is a Hermitian symmetric matrix, so it is sufficient to only consier the elements k,l with l k. The elements of A H A are given by A H A 1 l k k,l = s m+l k s m m= l k,k =,..., L 1 17 where s m are the elements of the vector s, efine in 3. Averaging those elements over the unknown ata symbols yiels for the elements of R A M+ν 1 E s nml k j2π R A k,l = l + k e m= l k,k =,..., L 1 18 REFERECES [1] D. Lanström, S.K. Wilson, J.-J. van e Beek; P. Öling an P.O. Börjesson. "Symbol Time Offset Estimation in Coherent OFDM Systems". IEEE Trans. Comm., 54:pp , April 22. [2] J. A. C. Bingham. 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